Question

Sol. [Key Concept: Compute $2a + 3b$ and $a - b$, then the triangle area $= \frac{1}{2}|(2a + 3b) \times (a - b)|$; square it.]

$${\ 2\overrightarrow{a} + 3\overrightarrow{b} = 2(2\widehat{i} + 3\widehat{j} + 3\widehat{k}) + 3(6\widehat{i} + 3\widehat{j} + 3\widehat{k}) }{\Rightarrow 22\widehat{i} + 15\widehat{j} + 15\widehat{k} }{\overrightarrow{a} - \overrightarrow{b} = - 4\widehat{i} }$$Area $= \frac{1}{2}|(2\overrightarrow{a} + 3\overrightarrow{b}) \times (\overrightarrow{a} - \overrightarrow{b})|$

$${= \frac{1}{2}| - 60\widehat{j} + 60\widehat{k}| = \frac{1}{2}\sqrt{(60)^{2} \times 2} }{A = \frac{60}{\sqrt{2}} }$$Square of area is $A^{2} = 1800$